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\begin_body
\begin_layout Title
Eclipse computations in nyx
\end_layout
\begin_layout Author
Christopher Rabotin
\end_layout
\begin_layout Standard
This computation assumes that all objects are spherical.
\end_layout
\begin_layout Standard
\begin_inset CommandInset toc
LatexCommand tableofcontents
\end_inset
\end_layout
\begin_layout Section
Summary
\end_layout
\begin_layout Standard
In short, we project the light source onto a plane which crosses the eclipsing
body perpendicularly to the direction between the spacecraft and the (potential
ly) eclipsing body.
We then find whether the disks representing the eclipsing body and the
light source on that plane overlap.
If they do not, then we're in full illumination.
If they do, we compute the overlapping area of both disks, and compute
the nominal apparent disk.
The ratio of these areas is used to compute the percentage of penumbra.
\end_layout
\begin_layout Section
Derivation
\end_layout
\begin_layout Subsection
Umbra or visibilis computation
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status collapsed
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename eclipse_computation.jpg
width 25col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
Top-view of eclipsing problem
\end_layout
\end_inset
\end_layout
\end_inset
\end_layout
\begin_layout Standard
First, let's compute
\begin_inset Formula $\beta_{2}$
\end_inset
the angle between
\begin_inset Formula $\vec{r}_{\text{EB-LS}}$
\end_inset
and
\begin_inset Formula $\vec{r}_{\text{LS}}$
\end_inset
, respectively the vector from the eclipsing body to the light source and
the vector from the spacecraft to the light source.
If that angle is less than a right angle, then the light source is not
behind the eclipsing body, so we're in full illumination:
\emph on
visibilis
\emph default
.
\end_layout
\begin_layout Standard
Then, we compute
\begin_inset Formula $\beta_{3}$
\end_inset
the angle between the spacecraft and the eclipsing body, and between the
spacecraft and the light source.
We need this to project the radius of the light source onto the plane centered
at the eclipsing geoid, and normal to the direction to the spacecraft.
\end_layout
\begin_layout Standard
Using the triangle formed between the spacecraft, the center of the eclipsing
body, and the projection plane, we can compute
\begin_inset Formula $\vec{r^{\prime}}_{\text{LS}}$
\end_inset
, the vector from the spacecraft to the intersection point on the plane
in the direction of
\begin_inset Formula $\hat{\boldsymbol{r}}_{\text{LS}}$
\end_inset
.
We use
\begin_inset Formula $\beta_{3}$
\end_inset
for this computation as we know the length of that hypotenuse is
\begin_inset Formula
\[
|\vec{r}_{\text{EB-LS}}|=|\vec{r^{\prime}}_{\text{LS}}|\cos\beta_{3}
\]
\end_inset
\end_layout
\begin_layout Standard
Using Thales' theorem, we can compute the
\begin_inset Quotes eld
\end_inset
pseudo light source radius
\begin_inset Quotes erd
\end_inset
, that is the radius of the light source as seen from an angle
\begin_inset Formula $\gamma$
\end_inset
from the plane.
Forming a triangle between the intersection point, the center of the eclipsing
body, and the orthogonal projection of that intersection point onto
\begin_inset Formula $\vec{r}_{\text{EB-LS}}$
\end_inset
, we can compute the actual radius of the light source on the projection
plane.
\end_layout
\begin_layout Standard
Let
\begin_inset Formula $\vec{r}_{\text{Plane-LS}}$
\end_inset
be the vector from the center of the eclipsing body to the projected center
of the light source on the plane.
We now check for any overlap.
If the norm of
\begin_inset Formula $\vec{r}_{\text{Plane-LS}}$
\end_inset
minus the project radius is greater than the radius of the eclipsing body,
it means that, no matter what the direction is, the shaddow of the eclipsing
body
\emph on
ends
\emph default
before the closest point of the light source: the light source is fully
visible.
If the norm of
\begin_inset Formula $\vec{r}_{\text{Plane-LS}}$
\end_inset
plus the projected radius of the light source is less than the radius of
the eclipsing body, then the light source is fully behind the eclipsing
body, so we're in total eclipse (
\emph on
umbra
\emph default
).
Note that we have ruled out the light source being in front of the eclipsing
body by computing the angle
\begin_inset Formula $\beta_{2}$
\end_inset
at the start.
\end_layout
\begin_layout Subsection
Penumbra percentage
\end_layout
\begin_layout Standard
\begin_inset Float figure
placement h
wide false
sideways false
status collapsed
\begin_layout Plain Layout
\align center
\begin_inset Graphics
filename eclipse_penumbra.jpg
width 25col%
\end_inset
\end_layout
\begin_layout Plain Layout
\begin_inset Caption Standard
\begin_layout Plain Layout
In plane view
\end_layout
\end_inset
\end_layout
\begin_layout Plain Layout
\end_layout
\end_inset
\end_layout
\begin_layout Standard
Both circles represent the light source and the eclipsing body.
We then use the
\begin_inset CommandInset href
LatexCommand href
name "Circle-Circle intersection"
target "http://mathworld.wolfram.com/Circle-CircleIntersection.html"
literal "false"
\end_inset
computation to compute the area of the asymmetric lens corresponding to
the overlap of both circles,
\begin_inset Formula $A_{\text{shadow}}$
\end_inset
.
Then, we compute the full area of the light source,
\begin_inset Formula $\mathcal{A_{\text{LS}}}$
\end_inset
.
\end_layout
\begin_layout Standard
The penumbra value,
\begin_inset Formula $P$
\end_inset
, is such that a number close to one means that the light source is almost
in full visibility.
Conversely, if the number is close to zero, then we are near total umbra.
\end_layout
\begin_layout Standard
\begin_inset Formula
\[
P=\frac{\mathcal{A_{\text{LS}}}-\mathcal{A_{\text{shadow}}}}{\mathcal{A_{\text{LS}}}}
\]
\end_inset
\end_layout
\end_body
\end_document
